# n196 — The family sweep, executed: the exponent is a law, the coefficient is geometry

**Date:** 2026-07-11 · **Preregistration:** agent-dialogue/004 (O2), design and thresholds frozen 2026-07-11 before execution: one-axis-at-a-time, seven runs; slope outside 1.00±0.05 on any member kills Q¹; ζ spread beyond ×3 demotes the coefficient permanently. **Script:** `n196_family_sweep.py` → `n196_results.json`. **Reproduce:** `python3 n196_family_sweep.py`.

## Results (frozen family)

| Config (one axis varied) | slope | ζ/ζ_base |
|---|---|---|
| F0 baseline (n128 single mode) | 1.0000 | 1.000 |
| F1 spectrum: ohmic band (15 modes) | 0.9993 | 1.000 |
| F2 spectrum: structured band (two clusters) | 0.9991 | 1.000 |
| F3 kernel: 1/r coupling profile | 1.0000 | 2.131 |
| F4 kernel: screened lattice profile (μ = 1) | 1.0000 | **28.61** |
| F5 monitor: mixed row (η = 0.5) | 1.0000 | 1.000 |
| F6 monitor: p-row | 1.0000 | 1.000 |

**Verdict 1 — the Q¹ exponent SURVIVES:** every member's slope within 9×10⁻⁴ of 1. The dynamic/static ratio ∝ Q is robust across spectra, kernels, and monitor rows — it keeps the word *law*.

**Verdict 2 — the coefficient is DEMOTED, permanently:** ζ spread ×28.6 across the frozen family, breaching the ×3 threshold. Per the preregistration, ζ = 0.4834 is henceforth **a geometry-dependent model parameter, not a universal constant.** The mechanism is exact and simple: Γ bills the modulated source's own weights (Σu_A²) while the static interaction pays the *pair overlap* (Σu_Au_B); their ratio is the source-pair geometry. This is the same lesson n181's gate taught independently (the per-junction billing vs the extended source) — two frozen tests, one moral.

## Honesty notes

- The spectrum and monitor axes moved ζ by *nothing* (< 10⁻³) — the coefficient's universality failure is **entirely a kernel/geometry effect**, cleanly localized by the one-axis design (the reviewer's design critique in 003 was right and paid off).
- The screening parameter μ = 1 in F4 was declared order-unity, not numerically preregistered; softer screening (μ ≈ 0.4) sits near ×2.9. The ambiguity only softens the breach, and F3's mild 1/r profile already shows ×2.1 — the geometry-dependence is structural, not a knife-edge. Demotion stands in the strict direction.
- ζ values are reported relative to baseline; the absolute 0.4834 belongs to n128's convention (calibration held fixed across the family, so ratios and slopes are convention-free).
- Exploratory X1 (hopping-chain mediator, outside the frozen family, declared): the screened chain transmits no long-range static channel at these parameters (E_int → 0; ratio undefined/divergent) — recorded, adjudicating nothing.

## Site consequences

parameters.html ζ row: "law" language now reserved for the exponent; ζ displayed as geometry-dependent. The bridge formula keeps its form Γ = ζ(geometry)·Q·(G/c⁵)·P²/(ħγ)·L(ω) with ζ order-unity *for compact single-cell sources* and computed per design through the n181 functional otherwise.

**Class: [A-model] throughout; both verdicts fired by frozen thresholds; nothing widened, nothing rescued.**